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ANALYSIS AND MITIGATION OF RISKS IN INFRASTRUCTURES | INFRARISK- !
Gidewon Goitom Tekeste Laboratório Nacional de Engenharia Civil (LNEC) and University of Aveiro (UA)
July 28
INFRARISK- PhD Programme – 3rd Summer Workshop
STABILITY ANALYSIS OF A SHAKE TABLE HYBRID SIMULATION FOR LINEAR AND NON-LINEAR SDOF SYSTEMS
2 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
OUTLINE
q Virtual hybrid simulation (VHS)
q Mechanical and complete system identification of LNEC’s uniaxial shake table (ST1D)
q ST1D under payload and uncertainty of its parameters
q Stability analysis of a VHS using Routh-Hurwitz stability test
q Stability test on linear and non-linear SDOFs
q VHS using OpenSees & OpenFresco (OSOF-VHS)
q Modelling experimental errors in OSOF-VHS
q Future directions
3 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Virtual hybrid simulation as a pre-hybrid testing for:
ü Checking bugs in the testing software
ü Selecting appropriate controllers
ü Addressing experimental errors and stability test
v Requirements of VHS: ü Complete model of the
servo-hydraulic actuator or the shake table
ü Model of the experimental element
Virtual Hybrid Simulation (VHS)
SIMULINK/LabVIEW
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1. Mechanical system identification Includes identification of: ü Total moving mass
§ Experimental sub-structure mass § Platen mass § Actuator mass § Other moving parts
ü Dissipative force § Experimental sub-structure damping § Resistance in actuator chamber § Resistance in platen bearings
ü Elastic force § Experimental sub-structure § Other sources
2. Complete system identification Includes: ü Mechanical system identification ü Servo-hydraulic actuator control
(SHAC) identification
Uniaxial shake table system identification
Servo-valve
Accumulators
Actuator
Accelerometer
Servo-controller Pump control
LVDT
Load cell
3m
2m
5 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Ofs_1=Ofs_2
v Mathematical models assumed for fitting:
ü F↓I =M u ↓x , where M= M↓Platen + M↓actuator ü F↓E =K u↓x ü F↓D =[F↓µ +C|u ↓x |↑α ]sign( u ↓x ), where 0≤α≤1
v The experimental method uses: ü Sinusoidal and triangular displacement signals
v Uses periodicity of measured signals: ü Triangular test:
ü Sinusoidal test:
Mechanical system identification
F↓E (u↓1, u↓2 )= 1/2 [F↓A (t↓1 )+ F↓A (t↓2 )]=K u↓x u↓x (t↓1 , t↓2 )= 1/2 [u↓x (t↓1 )+ u↓x (t↓2 )] (0< t↓1 <T) u↓x (t↓1 )= u↓x (t↓2 ), u ↓x (t↓1 )= − u ↓x (t↓2 ) & u ↓x (t↓1 )≈0, u ↓x (t↓2 )≈0
F↓I (u ↓1 , u ↓2 )= 1/2 [F↓A (t↓1 )+ F↓A (t↓2 )]=M u ↓x (t↓1 )
u ↓x (t↓1 , t↓2 )= 1/2 [u ↓x (t↓1 )+ u ↓x (t↓2 )] (0< t↓1 <T)
u ↓x (t↓1 )≈− u ↓x (t↓2 ) & u ↓x (t↓1 )≈u ↓x (t↓1 )
Ofs_1<Ofs_2
6 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Elastic force estimation: estimated from triangular signals v A low-pass Fourier filter with a fc 2-4 times the fcmd v Uses a synthesized velocity signal (from accelerometer and LVDT measurements) v Signals used for estimation:
v Null or near zero elastic force v Spurious estimates due to small SNR of the load cell force
Mechanical system identification continued…
Test Freq [Hz] Amplitude [cm] Vmax[cm/s]
T5 0.4 1.0 1.6
T6 0.5 1.0 2
same applies to mass and dissipative force estimation
7 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
𝑚↓𝑎𝑣𝑔 =2.063𝑡
Effective horizontal mass estimation: estimated from sinusoidal signals
v Excludes velocities below |±5|𝑚𝑚/𝑠 v Minimizing velocity sum at t1 and [T/2-t1-Tol, T/2-t1+Tol] v Test signals and estimates: v Weighting estimated through a simplified approach:
𝑤↓𝑖 = 𝑓↓𝑖 ∕∑↑▒𝑓↓𝑖 , 𝑚↓𝑎𝑣𝑔 =∑𝑖=1↑𝑛▒(𝑤↓𝑖 ∗𝑚↓𝑖 ) , where 𝑓↓𝑖 and 𝑚↓𝑖 are frequency and mass estimates of each test respectively. (a deduction from Lanese et al., 2012)
Mechanical system identification continued…
Test Freq [Hz]
Amplitude [cm] Vmax[cm/s] Amax [%g] Mass estimates
[ton] S1 0.2 4.0 5.0265 0.6435 1.98 S2 0.4 4.0 10.1300 2.5760 2.11 S3 0.5 4.0 12.5660 4.0240 2.06
u ↓min ( 𝑡↓1 )=min { [u ↓x (t↓1 )+ u ↓x (⏟T/2 − t↓𝟏 ┬t↓𝟐 ±Tol)] }
8 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Dissipative force estimation: estimated from triangular signals v Straightforward coupling of samples (no displacement sum minimization) v Propagation of disturbance is pronounced after inversion of motion v Each triangular test yield +ve and –ve estimates of dissipative force v The coulomb friction force governs the dissipative force
Mechanical system identification continued…
9 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Involves modelling: ü Servo-controller - proportional gain only controller ü Servo-valve - first order transfer function (good in 0-50Hz range) ü Hydraulic actuator - linearized flow equation and oil-column frequency ü Estimated effective mass of the platen (elastic force was set to zero) ü Dissipative force (simplified to viscous damping only) ü Payload (specimen) dynamic properties - no payload condition to produce a generalized
system model; Model subject to changes under any experimental specimen
Complete system identification
G↓x↓p u↓c = Ak↓p k↓sv k↓q /k↓pl /( A↑2 /k↓pl K↓h m↓T↑∗ τ↓sv ) s↑4 + {m↓T↑∗ τ↓sv + A↑2 /k↓pl K↓h (m↓T↑∗ +τ↓sv c↓t )} s↑3 + { τ↓sv (c↓t + A↑2 /k↓pl )+ m↓T↑∗ + c↓t A↑2 /k↓pl K↓h }s↑2
+ (c↓t + A↑2 /k↓pl )s+ Ak↓p k↓sv k↓q /k↓pl
Where: k↓pl = K↓c + C↓l , K↓h = 4β↓e A↑2 /V↓t and 𝒎↓𝑻↑∗ = 𝒎↓𝒑 + 𝒎↓𝒔𝒑 (𝟏+ 𝑯↓𝒔𝒑 ),andH↓sp (s)= − m↓sp s↑2 /m↓sp s↑2 +sc↓sp + k↓sp
v 𝒎↓𝑻↑∗ = 𝒎↓𝒑 under no payload condition, while, under payload 𝒎↓𝒔𝒑 and H↓sp (s) are non-zero terms
10 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Test signal characteristics: ü Band Limited White Noise (BLWN) ranging 0-50Hz ü RMS value of 0.345 cm
v Parametric identification of the ST1D model in SIMULINK using: ü Transfer system equation developed ü Command signal, measured signal, and the gain error identified ü Constrained non-linear least square solution
v Validation of estimated shake table parameters
Complete system identification continued…
Oilbulkmodulus(𝛽↓𝑒 )
Servo-valve@me
constant( 𝜏↓𝑠𝑣 ) Valvepressuregain+Leakagefactor ( 𝑘↓𝑝𝑙 )
193716.28Kpa 0.024641sec 1.67401e-7m3/s/Kpa
𝑯↓𝑺𝑻𝟏𝑫 = 𝟏.𝟗𝟒𝟖𝟓𝒆+𝟕/𝒔↑𝟒 + 𝟗𝟐.𝟐𝟗𝒔↑𝟑 +𝟐.𝟓𝟏𝟑𝟐𝒆+𝟒 𝒔↑𝟐 +𝟗.𝟑𝟒𝟕𝟖𝒆+𝟓𝒔+𝟏.𝟖𝟕𝟒𝟔𝒆+𝟕
(4poleswithoutzeros)
Op. freq
11 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
SDOF payload : Flexible and rigid v Flexible SDOF comes with Control-structure interaction
(CSI) ü Behaves similarly to a no-payload condition,
except it introduces: Ø A notch and a peak in the magnitude plot Ø A notch in the phase plot
v Rigid SDOF results in a stiffer response and a significant shift in the oil-column resonance
Uncertainty of ST1D parameters: No payload v Undershooting mass results in a spongy response v Overshooting apparently makes system stiffer v Increase in Kp: stiffer but smaller margins of stability
ST1D model under payload & uncertainty of its parameters
𝑓↓𝑛 =24.08 𝐻𝑧𝑓↓𝑆𝐷𝑂𝐹 =5𝐻𝑧
Kp,crt=10.529
12 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Stability analysis of a VHS using Routh-Hurwitz test
Why and how stability test: v Delay in HS is interpreted as a ‘negative damping’ v Stability under a linear transfer system is dictated by:
ü Experimental stiffness ü Experimental mass and ü Experimental damping
v Shortcomings of Mercan & Ricles (2007) stability study: ü A ‘pure delay’ assumption in HS
v System delay is a function of frequency, hence it requires the system model
v Stability test using Routh-Hurwitz stability: ü Sufficient for stability of LTI control systems ü Allows parametric study under a linear response
Routh-Hurwitz method: v Finds roots of the characteristic polynomial equation (D(s))
that fall in right-half S-plane v Procedure:
ü Develop the Hurwitz matrix (upper triangular matrix) ü Any sign change in 1st column indicates unstable
test
𝐺↓𝑉𝐻𝑆 (𝑠)= 𝑋↓𝑡 (𝑠)/𝑃(𝑠) = 𝑁(𝑠)/𝐷(𝑠)
𝑠↑𝑚
𝑏↓𝑚
𝑏↓𝑚
−2
𝑏↓𝑚
−4
𝑏↓𝑚
−6 …
𝑠↑𝑚
−1
𝑏↓𝑚
−1
𝑏↓𝑚
−3
𝑏↓𝑚
−5
𝑏↓𝑚
−7 …
𝑠↑𝑚
−2 𝑐↓1 𝑐↓2 𝑐↓3 𝑐↓4 …
𝑠↑𝑚
−3
𝑑↓1
𝑑↓2
𝑑↓3
𝑑↓4
…
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
𝑠↑0 𝑒↓1
Where: 𝒄↓𝒊 = 𝒃↓𝒎−𝒊 ∗𝒃↓𝒎−𝟐𝒊 − 𝒃↓𝒎 ∗𝒃↓𝒎−𝟐𝒊−𝟏 /𝒃↓𝒎−𝟏 and so on
Max: 6 poles & 2 zeros
𝐷(𝑠)= 𝑏↓𝑚 𝑠↑𝑚 + 𝑏↓𝑚−1 𝑠↑𝑚−1 + 𝑏↓𝑚−2 𝑠↑𝑚−2 +…=0
13 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Analysis and results: v A SDOF properties: ω=1Hz, m=2t and ζ=2% v Generating stability surface by constantly changing the fractions of
experimental and numerical sub-structures, e.g., Mexp= Mexp/m v A percentage step of 5% was adopted for each parameter v 2-parameter study, namely:
i. Kexp versus Cexp at: q Mexp= 0, 20%, 40%, 60%, 80%, 100%
ii. Cexp versus Mexp at: q Kexp= 0, 20%, 40%, 60%, 80%, 100%
iii. Kexp versus Mexp at: q Cexp= 0, 20%, 40%, 60%, 80%, 100%
v 2-parameter study at a fixed third parameter, but at varying values of stiffness and viscous damping of the SDOF (for cases i and ii) q m=2t (constant), ω=1,2,5 Hz, and ζ=2%, 5%, 10%
Stability analysis of a linear SDOF
14 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Kexp versus Cexp at constant Mexp: v As Mexp increases (0-80%), Cexp falls v As Mexp increases (0-40%), Kexp rises, but falls
when Mexp increases further v At full experimental inertial mass, system
attains improved stability v At 20% of Mexp, as ω and ζ of the SDOF
increase (constant mass),Kexp â, while Cexp á
v Validation of stability test at a transition zone, using El Centro at 0.3g PGA and a free-vibration tail, at Mexp=20%
Stability analysis of a linear SDOF
Mexp=0%
Mexp=20%
Mexp=100%Mexp=40%,
Mexp=80%
Mexp=60%
Damping
[Cexp]
S9ffness[Kexp]
15 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Cexp versus Mexp at constant Kexp: v Increasing Kexp, Cexp falls and the minimum
required Mexp rises v Mexp needs to be bounded for stability v At full experimental mass, stability is attained
if the minimum Cexp is increased v When ω and ζ of the SDOF increase
(constant mass), system exhibits a complex pattern
v Validation of stability test at Kexp=20%
Stability analysis of a linear SDOF continued…
Kexp=0% Kexp=20% Kexp=40%
Kexp= 60% Kexp =80% Kexp =100%
Mass[Mexp]
Damping
[Cexp]
16 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Kexp versus Mexp at varying Cexp: ü Regardless of the Cexp, stability contour is identical, except at Cexp=100% ü A positive linear relationship exists between Mexp and Kexp until 30% of Mexp ü Stable at right top corner of contour plot at Cexp=100%, i.e., shake table test ü Small deviations of the stable zone under varying frequencies and damping, but
the linear relationship still prevails
Stability analysis of a linear SDOF continued…
Tekeste,G.G.,Correia,A.A.,Costa,A.G.,[2017],“Stabilityanalysisofareal->meshaketablehybridsimula>onforlinearandnon-linearSDOFsystems”,7thinterna>onalconferenceonAdvancesinExperimentalStructuralEngineering,EUCENTREFounda>on,Pavia,Italy
17 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
Why? v To address the nonlinear range of response v Study the relation between degree of nonlinearity and
stability Methodology: v Simulink model of VHS
ü Bouc-Wen model of the experimental part ü Linear model of numerical part
𝒇↓𝒆𝒙𝒑 (𝐭)=𝛂 𝒌↓𝒊𝒏𝒕↑𝒆𝒙𝒑 𝒙↓𝒔𝒑 (𝐭)+(𝟏−𝜶)𝒌↓𝒊𝒏𝒕↑𝒆𝒙𝒑 𝜹↓𝒚 𝐙(𝐭)+ 𝒄↑𝒆𝒙𝒑 𝒙 ↓𝒔𝒑 (𝐭) 𝒁 (𝐭)= 𝒙 ↓𝒔𝒑 (𝐭){𝐀+𝛃∗𝐬𝐢𝐠𝐧(𝐙(𝒕)𝒙 ↓𝒔𝒑 (𝒕))+𝛄|𝐙(𝐭)|↑𝒏 }∕𝜹↓𝒚 v The Simulink model calls a MATLAB .m function of Bouc-
Wen at every step Comparison: v Validation at unstable coordinates of the linear case for
comparison (Mexp=20%, Cexp=20% and Kexp=35%) v Comments:
ü Improved stability under a non-linear response and/or a material with high non-linear behaviour
ü A larger margin from instability with increasing degree of non-linearity, defined by
𝝆=𝒎𝒂𝒙|𝒇↓𝒓 (𝒕)|/𝒇↓𝒎𝒂𝒙 = |𝒇↓𝒓 (𝒕)|/𝒇↓𝒚 [𝑨∕(𝜷+𝜸)] ↑𝟏/𝒏
Stability analysis of a nonlinear SDOF
18 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
OpenFresco architecture (credit to Schellenberg et al., 2009)
v Virtual hybrid tests using OS as the computational driver and OF as the middleware
v Restoring forces returned to OS are simulated using OS uniaxial material (restoring forces in actual hybrid tests are measured in laboratory)
Experimental errors modelling in OF: v Introducing experimental errors through
ExpSignalFilter control object using: v ErrorSimUndershoot v ErrorSimOvershoot v ErrorSimRandomGauss (models random
process in nature
Virtual hybrid simulation using OpenSees and OpenFresco
Tekeste,G.G.,Correia,A.A.,Costa,A.G.,[2017]“Virtualhybridsimula>ontestsaccoun>ngforexperimentalerrors”,OpenSeesDaysEurope:FirstEuropeanconferenceonOpenSees,Porto,Portugal
19 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Offline estimation of error parameters using the ST1D model in Simulink
Modelling experimental errors in OSOF-VHS
ML method
#expSignalFilterErrorSimRandomGauss$tag$avg$std expSignalFilterErrorSimRandomGauss1-0.00034750.2051730v One-bay frame with a truss element and two non-linear columns v Columns are “experimental” substructures v Truss element and masses are the “numerical” substructures
20 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v OpenFresco definition: expControlSimUniaxialMaterials11-ctrlFilters10000expSetupOneActuator1-control12-sizeTrialOut33 (only 1 ctrl and 1 out were used) expSiteLocalSite11 expElementbeamColumn1131-site1–initStif...v Since “real” experimental errors cannot be directly feedback to OS, a number of stochastic
realizations were necessary to model the errors using WGN v Expected response is found by averaging over 50 realizations v The sub-space synchronization plot (SSP) shows both gain and time-lag errors
Modelling experimental errors in OSOF-VHS continued…
Coupling force returned to OS Trial displacements without error vs with WGN (averaged)
21 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Conclusions: ü Experimental errors result in a coupled gain and delay (introduced by ST1D
dynamics) ü Reduction in energy dissipation capacity is prevalent
v Solutions sought: v Higher frequency of AD/DA conversion v Tuning the controller gains for minimal settling time and overshoot v Implementation of adaptive model based compensator in the outer loop
Modelling experimental errors in OSOF-VHS continued…
Column hysteresis of VHS with & without Exp. errors
Feedforward compensation of ST1D
22 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
v Completion of a LabVIEW based testing software that works in conjunction with OF and OS
v Conducting hybrid simulations on a steel frame using the ST1D only
v Development of advanced control strategies for the RTHS framework
ü Shake table control
ü Force control of actuators
ü Model based and adaptive compensation
Future directions
23 Gidewon G.Tekeste / Stability analysis of a shake table hybrid simulation for linear and non-linear SDOF systems
THANK YOU FOR YOUR ATTENTION! Gidewon G.Tekeste ([email protected])